1. If the surface area of a cube is 294 inches squared, find the volume.
2. If the base of a prism is an equilateral triangle with sides 8cm, and each lateral edge measures 10cm, what is the volume of the prism.
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1. To find the volume of a cube when given its surface area, we can use the equation V = s^3, where s is the length of one side of the cube.
In this case, the surface area of the cube is given as 294 inches squared. Let's assume that s represents the length of one side of the cube.
To find s, we can use the formula for the surface area of a cube: SA = 6s^2.
Given that SA = 294 inches squared, we can solve for s:
6s^2 = 294
Divide both sides by 6:
s^2 = 49
Take the square root of both sides:
s = 7
Now that we have the length of one side of the cube, we can calculate the volume:
V = s^3 = 7^3 = 343 cubic inches.
Therefore, the volume of the cube is 343 cubic inches.
2. To find the volume V of a prism when given the base area A and the height h, we can use the formula V = A * h.
In this case, the base of the prism is an equilateral triangle with sides measuring 8 cm, and each lateral edge measures 10 cm. Let's find the area of the base first.
The area A of an equilateral triangle can be calculated using the formula A = (√3 / 4) * s^2, where s is the length of one side of the triangle.
Given that s = 8 cm, we can substitute it into the formula:
A = (√3 / 4) * (8)^2
A = (√3 / 4) * 64
A = (√3 / 4) * 64
A = (√3 * 64) / 4
A = 16√3 cm^2 (approximated)
Now that we have the base area A and the height h, we can calculate the volume:
V = A * h = 16√3 cm^2 * 10 cm
V = 160√3 cm^3 (approximated)
Therefore, the volume of the prism is approximately 160√3 cm^3.
To find the volume of a cube, we need to know its side length. However, in this problem, we are given the surface area of the cube. To find the side length, we can use the formula for the surface area of a cube, which is given by:
Surface Area = 6 * (Side Length)^2
In this case, the surface area given is 294 inches squared. We can set up the equation as follows:
294 = 6 * (Side Length)^2
To isolate the side length, we first divide both sides of the equation by 6:
294/6 = (Side Length)^2
49 = (Side Length)^2
To find the side length, we take the square root of both sides of the equation:
√49 = √(Side Length)^2
7 = Side Length
Now that we know the side length of the cube is 7 inches, we can calculate the volume. The volume of a cube is given by:
Volume = (Side Length)^3
Substituting the value of the side length, we get:
Volume = 7^3 = 7 * 7 * 7 = 343 cubic inches
Therefore, the volume of the cube is 343 cubic inches.
Now let's move on to the second question about the prism:
The prism in question has an equilateral triangle as its base with sides measuring 8 cm. The lateral edges of the prism measure 10 cm each. To find the volume of the prism, we can use the formula:
Volume = (Base Area) * Height
The base of the prism is an equilateral triangle, and the formula for the area of an equilateral triangle is given by:
Base Area = (sqrt(3) / 4) * (Side Length)^2
Substituting the given side length, we get:
Base Area = (sqrt(3) / 4) * (8 cm)^2
= (sqrt(3) / 4) * 64 cm^2
= (8 * sqrt(3)) cm^2
Now, we need to find the height of the prism. Since the lateral edges of the prism measure 10 cm each, the height can be calculated by using the Pythagorean theorem. In a right-angled triangle, the hypotenuse (lateral edge in this case) is related to the height and the base (side of the equilateral triangle) as follows:
(lateral edge)^2 = (base)^2 + (height)^2
Substituting the given values, we get:
10^2 = 8^2 + (height)^2
100 = 64 + (height)^2
(height)^2 = 100 - 64
(height)^2 = 36
height = sqrt(36) = 6 cm
Now that we know the base area (8*sqrt(3) cm^2) and the height (6 cm), we can calculate the volume:
Volume = (Base Area) * Height
= (8 * sqrt(3)) cm^2 * 6 cm
= 48 * sqrt(3) cm^3
Therefore, the volume of the prism is 48 * sqrt(3) cubic centimeters.
A cube has 6 sides. Divide 294 by 6 to get the area of 1 side. Take the square root to get the length of a side.
Volume is s^3.